1: \(x^3+x^2+x+1\)
\(=x^2\left(x+1\right)+\left(x+1\right)\)
\(=\left(x+1\right)\left(x^2+1\right)\)
2: \(x^2y+xy^2-x-y\)
=xy(x+y)-(x+y)
=(x+y)(xy-1)
3: \(x^2-xy+5x-5y\)
=x(x-y)+5(x-y)
=(x-y)(x+5)
4: \(2x^2-x-6xy+3y\)
=x(2x-1)-3y(2x-1)
=(2x-1)(x-3y)
5: \(x^2+7x+7y-y^2\)
\(=\left(x^2-y^2\right)+7\left(x+y\right)\)
=(x-y)(x+y)+7(x+y)
=(x+y)(x-y+7)
6: \(x^2-2x-9y^2+6y\)
\(=\left(x^2-9y^2\right)-2\left(x-3y\right)\)
=(x-3y)(x+3y)-2(x-3y)
=(x-3y)(x+3y-2)
7: \(x^2+2xy+y^2-25=\left(x+y\right)^2-5^2\)
=(x+y+5)(x+y-5)
8: \(x^2y^2-x^2+8x-16\)
\(=\left(xy\right)^2-\left(x^2-8x+16\right)\)
\(=\left(xy\right)^2-\left(x-4\right)^2\)
=(xy+x-4)(xy-x+4)
9: \(5x^2-5xy+7y-7x\)
=5x(x-y)-7(x-y)
=(x-y)(5x-7)
10: \(5x^2-5y^2-4\left(x+y\right)^2\)
\(=5\left(x^2-y^2\right)-4\left(x+y\right)^2\)
=(x+y)(5x-5y-4x-4y)
=(x+y)(x-9y)
11: \(a^2-ac+b^2+2ab-ac\)
\(=\left(a^2+2ab+b^2\right)-c\left(a+b\right)\)
\(=\left(a+b\right)^2-c\left(a+b\right)\)
=(a+b)(a+b-c)
12: \(a^2-ac-9b^2+3bc\)
=(a-3b)(a+3b)-c(a-3b)
=(a-3b)(a+3b-c)
13: \(x^2-6xy+9y^2+4x-12y\)
\(=\left(x-3y\right)^2+4\left(x-3y\right)\)
=(x-3y)(x-3y+4)
14: \(x^3-5x^2-5x+1\)
\(=\left(x^3+1\right)-5x\cdot\left(x+1\right)\)
\(=\left(x+1\right)\left(x^2-x+1\right)-5x\left(x+1\right)\)
\(=\left(x+1\right)\left(x^2-6x+1\right)\)
15: \(3x^3-x^2-21x+7\)
\(=x^2\left(3x-1\right)-7\left(3x-1\right)\)
\(=\left(3x-1\right)\left(x^2-7\right)\)
16: \(x^3-4x^2+8x-8\)
\(=\left(x^3-8\right)-4x\left(x-2\right)\)
\(=\left(x-2\right)\left(x^2+2x+4\right)-4x\left(x-2\right)\)
\(=\left(x-2\right)\left(x^2-2x+4\right)\)
